Group #21447865 — subgroups

Subgroups — #21447865

Order 7808.

166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14519722	C₂	1	2	yes	maximal
2	#16840142	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic
4	#16839567	C₂	1	2	yes	maximal, complemented
5	C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₄₄	—	2	1	no	—
7	C₂×Dic₂₄₄	C₂²	1	2	yes	—
8	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
9	#14519777	—	2	1	no	—
10	C₄×D₂₄₄	—	2	1	no	—
11	#14519722	—	2	4	no	—
12	#14519711	C₂²	1	1	yes	characteristic
13	#14519699	—	2	1	no	—
14	C₂×C₁₆×C₆₁	C₂²	1	2	yes	—
15	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×D₂₄₄	—	2	1	no	—
17	C₂×C₄×D₆₁	—	4	1	no	—
18	C₂×Dic₁₂₂	—	2	2	no	—
19	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
20	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
21	C₄²×C₆₁	D₄	1	1	yes	characteristic
22	#11758198	—	4	1	no	—
23	C₄⋊Dic₆₁	—	2	1	no	—
24	C₄⋊Dic₆₁	—	4	1	no	—
25	C₄⋊Dic₆₁	—	4	1	no	—
26	Dic₂₄₄	—	2	2	no	—
27	Dic₂₄₄	—	2	2	no	—
28	Dic₂₄₄	—	4	2	no	—
29	D₄₈₈	—	2	1	no	—
30	D₄₈₈	D₄	1	2	yes	—
31	C₁₆×C₆₁	—	2	2	no	—
32	C₂²×D₆₁	—	4	1	no	—
33	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
34	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₆₁	—	4	1	no	—
37	C₂×Dic₆₁	—	4	2	no	—
38	D₂₄₄	—	2	2	no	—
39	D₂₄₄	—	4	1	no	—
40	C₄×D₆₁	—	8	2	no	—
41	Dic₁₂₂	—	4	2	no	—
42	Dic₁₂₂	—	4	2	no	—
43	C₈×C₆₁	—	2	1	no	—
44	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₈×C₆₁	#41	1	1	yes	characteristic
46	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
47	C₂×D₆₁	—	4	2	no	—
48	C₂×D₆₁	—	8	1	no	—
49	C₂×D₆₁	—	8	1	no	—
50	C₄×C₆₁	—	2	1	no	—
51	C₄×C₆₁	SD₃₂	1	2	yes	complemented
52	C₄×C₆₁	#121	1	1	yes	characteristic
53	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
54	Dic₆₁	—	8	2	no	—
55	Dic₆₁	—	8	1	no	—
56	#2208	—	61	1	no	maximal, Sylow
57	C₂×C₆₁	#459	1	1	yes	characteristic
58	C₂×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
59	C₂×C₆₁	#510	1	1	yes	characteristic
60	D₆₁	—	8	2	no	—
61	C₂×SD₃₂	—	61	2	no	—
62	#494	—	61	1	no	—
63	C₄×D₈	—	61	1	no	—
64	C₄⋊C₁₆	D₆₁	1	1	yes	characteristic, complemented
65	#358	—	61	2	no	—
66	C₆₁	#2208	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₁₆	—	61	2	no	—
68	C₂×Q₁₆	—	122	1	no	—
69	C₂×D₈	—	61	1	no	—
70	C₄⋊Q₈	—	122	1	no	—
71	C₄×D₄	—	122	1	no	—
72	SD₃₂	—	122	4	no	—
73	C₂×C₁₆	C₂×D₆₁	1	2	yes	—
74	#107	—	61	1	no	—
75	#102	—	122	1	no	—
76	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
77	C₂×Q₈	—	122	2	no	—
78	C₂×D₄	—	122	1	no	—
79	C₂²×C₄	—	244	1	no	—
80	Q₁₆	—	122	2	no	—
81	Q₁₆	—	122	2	no	—
82	Q₁₆	—	244	2	no	—
83	D₈	—	61	2	no	—
84	D₈	—	122	1	no	—
85	C₂×C₈	D₂₄₄	1	1	yes	characteristic
86	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
87	C₄⋊C₄	—	122	1	no	—
88	C₄⋊C₄	—	244	1	no	—
89	C₄⋊C₄	—	244	1	no	—
90	C₂²⋊C₄	—	244	1	no	—
91	C₄²	D₂₄₄	1	1	yes	characteristic
92	C₁₆	—	2	2	no	—
93	C₂³	—	244	1	no	—
94	Q₈	—	244	2	no	—
95	Q₈	—	244	2	no	—
96	D₄	—	122	2	no	—
97	D₄	—	244	1	no	—
98	C₂×C₄	—	488	2	no	—
99	C₂×C₄	—	244	1	no	—
100	C₂×C₄	—	244	2	no	—
101	C₂×C₄	D₄₈₈	1	1	yes	characteristic
102	C₂×C₄	D₄₈₈	1	1	yes	characteristic
103	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
104	C₈	—	2	1	no	—
105	C₈	D₄×D₆₁	1	1	yes	characteristic
106	C₈	#11758226	1	1	yes	characteristic
107	C₂²	—	488	1	no	—
108	C₂²	—	488	1	no	—
109	C₂²	—	244	2	no	—
110	C₂²	C₂×D₄₈₈	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	488	2	no	—
113	C₄	—	488	1	no	—
114	C₄	#14519722	1	2	yes	complemented
115	C₄	#14519752	1	1	yes	characteristic
116	C₄	#14519805	1	1	yes	characteristic
117	C₂	—	488	2	no	—
118	C₂	#16839958	1	1	yes	characteristic
119	C₂	C₂×#14519722	1	1	yes	characteristic
120	C₂	#16840205	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519722

C₂

yes

maximal

#16840142

C₂

yes

maximal, characteristic, complemented

C₄×D₄₈₈

C₂

yes

maximal, characteristic

#16839567

C₂

yes

maximal, complemented

C₆₁×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

C₂²

yes

—

C₂×D₄₈₈

C₂²

yes

characteristic

#14519777

—

C₄×D₂₄₄

—

#14519722

—

#14519711

C₂²

yes

characteristic

#14519699

—

C₂×C₁₆×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₂₄₄

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

D₄₈₈

—

D₄₈₈

D₄

yes

—

C₁₆×C₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×D₄

yes

characteristic

C₈×C₆₁

#41

yes

characteristic

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

SD₃₂

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×SD₃₂

yes

characteristic

C₂×C₆₁

#510

yes

characteristic

D₆₁

—

C₂×SD₃₂

—

#494

—

C₄×D₈

—

C₄⋊C₁₆

D₆₁

yes

characteristic, complemented

#358

—

C₆₁

#2208

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

122

—

C₂×D₈

—

C₄⋊Q₈

—

122

—

C₄×D₄

—

122

—

SD₃₂

—

122

—

C₂×C₁₆

C₂×D₆₁

yes

—

#107

—

#102

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

122

—

C₂×D₄

—

122

—

C₂²×C₄

—

244

—

Q₁₆

—

122

—

Q₁₆

—

122

—

Q₁₆

—

244

—

D₈

—

D₈

—

122

—

C₂×C₈

D₂₄₄

yes

characteristic

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₂²⋊C₄

—

244

—

C₄²

D₂₄₄

yes

characteristic

C₁₆

—

C₂³

—

244

—

Q₈

—

244

—

Q₈

—

244

—

D₄

—

122

—

D₄

—

244

—

C₂×C₄

—

488

—

C₂×C₄

—

244

—

100

C₂×C₄

—

244

—

101

C₂×C₄

D₄₈₈

yes

characteristic

102

C₂×C₄

D₄₈₈

yes

characteristic

103

C₂×C₄

C₂×D₂₄₄

yes

characteristic

104

C₈

—

105

C₈

D₄×D₆₁

yes

characteristic

106

C₈

#11758226

yes

characteristic

107

C₂²

—

488

—

108

C₂²

—

488

—

109

C₂²

—

244

—

110

C₂²

C₂×D₄₈₈

yes

characteristic

111

C₄

—

112

C₄

—

488

—

113

C₄

—

488

—

114

C₄

#14519722

yes

complemented

115

C₄

#14519752

yes

characteristic

116

C₄

#14519805

yes

characteristic

117

C₂

—

488

—

118

C₂

#16839958

yes

characteristic

119

C₂

C₂×#14519722

yes

characteristic

120

C₂

#16840205

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₈₈
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×C₄⋊C₁₆	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	#2208
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₈₈

commutator

C₂×C₈×C₆₁

C₂³

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×C₄⋊C₁₆

C₂

socle

C₂²×C₆₁

C₂×D₈

hypercenter

C₄⋊C₁₆

D₆₁

hypocenter

C₆₁

#2208

LCS

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
2. #16840142	C₂	semidirect
4. #16839567	C₂	semidirect
5. C₆₁×C₄⋊C₁₆	C₂	semidirect
51. C₄×C₆₁	SD₃₂	semidirect
64. C₄⋊C₁₆	D₆₁	semidirect
66. C₆₁	#2208	semidirect
114. C₄	#14519722	semidirect

H ≅ G/N

type

2. #16840142

C₂

semidirect

4. #16839567

C₂

semidirect

5. C₆₁×C₄⋊C₁₆

C₂

semidirect

51. C₄×C₆₁

SD₃₂

semidirect

64. C₄⋊C₁₆

D₆₁

semidirect

66. C₆₁

#2208

semidirect

114. C₄

#14519722

semidirect

Subgroups — #21447865

Conjugacy classes of subgroups

Complements